Nonlinear Differential Equations

In the previous chapter, we developed techniques for solving first-order differential equations which are either linear or separable. However, in general the models we develop to describe biological systems will not be linear-indeed, we shall shortly see several types of biological systems which are driven by distinctly nonlinear interactions. Most nonlinear differential equations cannot be solved outright; we can, however, develop tools to analyze the long-term behavior of these models, as was done for difference equations in Chapter 2, and often the long-term behavior of the biological system is what really interests us. (If we are interested in short-term behavior of the system, we typically use computers to approximate the solution numerically.1)

In this chapter, we shall first develop these qualitative analysis tools for single first-order differential equations, taking care to use notation which will help us extend them to systems of differential equations in the next chapter. Following this, we shall study particular types of nonlinear interactions such as harvesting (last seen in Section 2.5) and contact processes. One of the fundamental results in biological modeling is the description of threshold quantities which control sharp changes in the nature of a system-for example, the difference between survival and extinction-and we shall also take time to examine common biological thresholds under the mathematical lens of bifurcations.